Convergences in W*-algebras

Abstract

Let U be a W*-algebra with a faithful normal state ρ{variant}; xn in U, n ≥ 1. Five conditions, each of which generalizes the notion of a.e. convergence for random variables, are studied with the use of some properties of the geometry of projectors from U. The conditions are equivalent if (xn) is bounded, and are not equivalent for an arbitrary sequence. Sequences xn → 0 a.u. yn → 0 a.u., xn + yn {not right arrow} 0 a.u. are given. Thus, some problems stated by Batty, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 48 (1979), 177-191, are solved. © 1986.